That conflates the centrists, including left and right (when there was a center-right) with the left. The progressive left was always dubious at best about globalization, iirc.
The migration thing is, to a great extent, people inflaming ethnic hatred. People like open borders when they want to migrate themselves, or hire migrants; it also brings trade and tourism; it brings peace and eliminates pointless political distinctions.
I agree that globalization needs to address the relatively easy mobility of capital relative to labor. But that problem is arguably extreme capitalism - where the capitalists get whatever they want - not globalization. Globalization writ by labor would have looked a lot different.